Solution
= Solution
The coordinate <partial derivatives> are $\xi_y=1/\sqrt y$ and $\eta_y=-1/\sqrt y$. The <multivariable chain rule> therefore gives
$$
u_y=\frac{U_\xi-U_\eta}{\sqrt y}.
$$
Since $\xi-\eta=4\sqrt y>0$, the expression entirely in the new coordinates is
$$
\boxed{u_y=\frac4{\xi-\eta}(U_\xi-U_\eta)}.
$$
The inverse coordinate formulas are $x=(\xi+\eta)/2$ and $y=(\xi-\eta)^2/16$, with domain $\xi>\eta$.